© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a158
K11a158
K11a160
K11a160
K11a159
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   The Knot K11a159

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Acknowledgement

K11a159 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,4,11,3 X18,6,19,5 X14,8,15,7 X2,10,3,9 X22,11,1,12 X8,14,9,13 X20,15,21,16 X6,18,7,17 X16,19,17,20 X12,21,13,22

Gauss Code: {1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -4, 8, -10, 9, -3, 10, -8, 11, -6}

DT (Dowker-Thistlethwaite) Code: 4 10 18 14 2 22 8 20 6 16 12

Alexander Polynomial: t-3 - 9t-2 + 27t-1 - 37 + 27t - 9t2 + t3

Conway Polynomial: 1 - 3z4 + z6

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {111, 2}

Jones Polynomial: q-3 - 3q-2 + 7q-1 - 11 + 15q - 18q2 + 18q3 - 15q4 + 12q5 - 7q6 + 3q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial: {K11a347, ...}

A2 (sl(3)) Invariant: q-10 - q-6 + 3q-4 - 2q-2 + 3q2 - 4q4 + 2q6 - 2q8 + q10 + 2q12 - 2q14 + 4q16 - q18 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial: - a-8 + 2a-6 + 3a-6z2 - a-4 - 4a-4z2 - 3a-4z4 + a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial: a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 3a-8z6 + 3a-7z3 - 7a-7z5 + 5a-7z7 - 2a-6 + 5a-6z2 - 4a-6z4 - 3a-6z6 + 5a-6z8 - a-5z + 9a-5z3 - 14a-5z5 + 5a-5z7 + 3a-5z9 - a-4 + 3a-4z2 - 11a-4z6 + 8a-4z8 + a-4z10 + a-3z + 4a-3z3 - 8a-3z5 - 3a-3z7 + 6a-3z9 - a-2 + 3a-2z2 + a-2z4 - 13a-2z6 + 7a-2z8 + a-2z10 + 6a-1z3 - 10a-1z5 + 3a-1z9 - 1 + 5z2 - z4 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {0, 3}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17           1
j = 15          2 
j = 13         51 
j = 11        72  
j = 9       85   
j = 7      107    
j = 5     88     
j = 3    710      
j = 1   59       
j = -1  26        
j = -3 15         
j = -5 2          
j = -71           


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
Show[DrawMorseLink[Knot[11, Alternating, 159]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 159]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 6, 19, 5], X[14, 8, 15, 7], 
 
>   X[2, 10, 3, 9], X[22, 11, 1, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], 
 
>   X[6, 18, 7, 17], X[16, 19, 17, 20], X[12, 21, 13, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 159]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -4, 8, -10, 9, -3, 10, 
 
>   -8, 11, -6]
In[5]:=
DTCode[Knot[11, Alternating, 159]]
Out[5]=   
DTCode[4, 10, 18, 14, 2, 22, 8, 20, 6, 16, 12]
In[6]:=
alex = Alexander[Knot[11, Alternating, 159]][t]
Out[6]=   
       -3   9    27             2    3
-37 + t   - -- + -- + 27 t - 9 t  + t
             2   t
            t
In[7]:=
Conway[Knot[11, Alternating, 159]][z]
Out[7]=   
       4    6
1 - 3 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 159]}
In[9]:=
{KnotDet[Knot[11, Alternating, 159]], KnotSignature[Knot[11, Alternating, 159]]}
Out[9]=   
{111, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 159]][q]
Out[10]=   
       -3   3    7              2       3       4       5      6      7    8
-11 + q   - -- + - + 15 q - 18 q  + 18 q  - 15 q  + 12 q  - 7 q  + 3 q  - q
             2   q
            q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 159], Knot[11, Alternating, 347]}
In[12]:=
A2Invariant[Knot[11, Alternating, 159]][q]
Out[12]=   
 -10    -6   3    2       2      4      6      8    10      12      14
q    - q   + -- - -- + 3 q  - 4 q  + 2 q  - 2 q  + q   + 2 q   - 2 q   + 
              4    2
             q    q
 
       16    18    20      22    24    26
>   4 q   - q   - q   + 2 q   - q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 159]][a, z]
Out[13]=   
                                           2      2      2
      -8   2     -4    -2    2      2   3 z    4 z    3 z     2  2      4
-1 - a   + -- - a   + a   + a  - 3 z  + ---- - ---- + ---- + a  z  - 2 z  - 
            6                             6      4      2
           a                             a      a      a
 
       4      4    6
    3 z    2 z    z
>   ---- + ---- + --
      4      2     2
     a      a     a
In[14]:=
Kauffman[Knot[11, Alternating, 159]][a, z]
Out[14]=   
                                                                2      2
      -8   2     -4    -2    2   z    z    z             2   3 z    5 z
-1 - a   - -- - a   - a   - a  + -- - -- + -- - a z + 5 z  + ---- + ---- + 
            6                     9    5    3                  8      6
           a                     a    a    a                  a      a
 
       2      2                3      3      3      3      3
    3 z    3 z       2  2   2 z    3 z    9 z    4 z    6 z         3    4
>   ---- + ---- + 3 a  z  - ---- + ---- + ---- + ---- + ---- + 6 a z  - z  - 
      4      2                9      7      5      3     a
     a      a                a      a      a      a
 
       4      4    4              5      5       5      5       5
    5 z    4 z    z       2  4   z    7 z    14 z    8 z    10 z         5
>   ---- - ---- + -- - 3 a  z  + -- - ---- - ----- - ---- - ----- - 8 a z  - 
      8      6     2              9     7      5       3      a
     a      a     a              a     a      a       a
 
              6      6       6       6              7      7      7
       6   3 z    3 z    11 z    13 z     2  6   5 z    5 z    3 z         7
>   7 z  + ---- - ---- - ----- - ----- + a  z  + ---- + ---- - ---- + 3 a z  + 
             8      6      4       2               7      5      3
            a      a      a       a               a      a      a
 
              8      8      8      9      9      9    10    10
       8   5 z    8 z    7 z    3 z    6 z    3 z    z     z
>   4 z  + ---- + ---- + ---- + ---- + ---- + ---- + --- + ---
             6      4      2      5      3     a      4     2
            a      a      a      a      a            a     a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 159]], Vassiliev[3][Knot[11, Alternating, 159]]}
Out[15]=   
{0, 3}
In[16]:=
Kh[Knot[11, Alternating, 159]][q, t]
Out[16]=   
         3     1       2       1       5      2      6    5 q       3
9 q + 7 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q  t + 
              7  4    5  3    3  3    3  2      2   q t    t
             q  t    q  t    q  t    q  t    q t
 
       5        5  2       7  2      7  3      9  3      9  4      11  4
>   8 q  t + 8 q  t  + 10 q  t  + 7 q  t  + 8 q  t  + 5 q  t  + 7 q   t  + 
 
       11  5      13  5    13  6      15  6    17  7
>   2 q   t  + 5 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a159
K11a158
K11a158
K11a160
K11a160